Nevermind the fact that these two words are both irregularly feminine
I was always taught in school that llave is key, like the kind you use to open doors / crank a vehicle, etcetera. Though when I learned one day that llave can also refer to the faucet on your sink or bathtub, I was, to say the least, confused. My first instinctive idea was to figure out the link between a door key and a water faucet, and it has continue to stump me.
Clave on the other hand seems to be the more abstract way to say key... the kind of key you would hear about in a project.
The key to understanding English is to expect anything.
The key to eating healthy is to eat mainly fresh vegetables.
I know that clave is also understood in the musical context.
This guitar is in the key of C minor. (@steverrill)
He likes to touch the keys on the piano.
I realize there could be regional differences with these words, but I am interested to see whether there is a deep, embedded meaning inside these words that allows their meanings to be used somewhat similarly. Because even though most people used the word contraseña for password, I also know that clave is used sometimes instead. And in my humble opinion, a password is pretty much the same as a key, well, more like a code, I suppose.

destonadoto reference an out of tune guitar before and been understood. – dockeryZ Mar 13 '15 at 11:59desafinada.fuera de tonogives the idea that it is out of key (though possibly in tune with itself.) – Level River St Mar 14 '15 at 15:30